The GW Approximation. Manish Jain 1. July 8, Department of Physics Indian Institute of Science Bangalore 1/36

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1 1/36 The GW Approximation Manish Jain 1 Department of Physics Indian Institute of Science Bangalore July 8, 2014

2 Ground-state properties 2/36 Properties that are intrinsic to a system with all its electrons in equilibrium. Density functional theory is the standard model for understanding ground-state properties. Total energy is a functional of the charge density. Kohn-Sham formulation: Map the interacting many-electron problem to non-interacting electrons moving in a self-consistent eld. ( V ionic(r) + V Hartree (r) + V xc (r) ) ψ(r) = ɛ ψ(r)

3 Ground-state properties 2/36 Properties that are intrinsic to a system with all its electrons in equilibrium. Density functional theory is the standard model for understanding ground-state properties. Total energy is a functional of the charge density. Kohn-Sham formulation: Map the interacting many-electron problem to non-interacting electrons moving in a self-consistent eld. ( V ionic(r) + V Hartree (r) + V xc (r) ) ψ(r) = ɛ ψ(r) Local density approximation Generalized gradient approximation

4 Ground-state properties 3/36 Crystal Structure 1 Phase transitions 1 Phonons 2 Charge density 1 1 M. T. Yin and M. L. Cohen, Phys. Rev. B 26, 5668 (1982). 2 P. Giannozzi, S. de Gironcoli, P. Pavone, and S. Baroni, Phys. Rev. B, 43, 7231 (1991).

5 Excited-state properties 4/36 Spectroscopic properties that involve experiments creating an excited particle above the ground state.

6 Excited-state properties 4/36 Spectroscopic properties that involve experiments creating an excited particle above the ground state. Light source Detector

7 Excited-state properties 4/36 Spectroscopic properties that involve experiments creating an excited particle above the ground state. Light source Detector ω

8 Excited-state properties 4/36 Spectroscopic properties that involve experiments creating an excited particle above the ground state. Light source Detector ω E k ε k hole Band structure: ε k = ω E k

9 Excited-state properties 4/36 Spectroscopic properties that involve experiments creating an excited particle above the ground state. Light source Detector ω E k ε k hole Band structure: ε k = ω E k

10 Excited-state properties - Quasiparticle gap 5/36 Direct Photoemission Inverse Photoemission e ω ω C C e V V E N 1 E N E N E N+1 Quasiparticle band gap dened as E g = Ionization Energy - Electron Anity = E N+1 + E N 1 2E N Dierent from Optical gap.

11 Ground-state methods for excited states 6/36 Kohn-Sham DFT works well for the ground state.

12 Ground-state methods for excited states 6/36 Kohn-Sham DFT works well for the ground state. Kohn-Sham eigenvalues are Lagrange multipliers.

13 Ground-state methods for excited states 6/36 Kohn-Sham DFT works well for the ground state. Kohn-Sham eigenvalues are Lagrange multipliers. Kohn-Sham eigenvalues eigenvalues of a ctive system.

14 Ground-state methods for excited states 6/36 Kohn-Sham DFT works well for the ground state. Kohn-Sham eigenvalues are Lagrange multipliers. Kohn-Sham eigenvalues eigenvalues of a ctive system. Kohn-Sham energies cannot be interpreted as removal/addition energies (except the energy of the highest occupied molecular orbital in a nite system).

15 Excited-state properties 7/36 Materials: InSb, InAs, Ge, GaSb, Si, InP, GaAs, CdS, AlSb, AlAs, CdSe, CdTe, BP, SiC, C 60, GaP, AlP, ZnTe, ZnSe, c-gan, w-gan, InS, w-bn, c-bn, diamond, w-aln, LiCl, LDA Fluorite, LiF Experimental Band Gap (ev) Theoretical Band Gap (ev) Quasiparticle Gap Non-interacting Experiment Energy (ev) Optical absorption 2 1 S. G. Louie in Topics in Computational Materials Science, edited by C. Y. Fong (World Scientic, Singapore, 1997). 2 J. Deslippe, G. Samsonidze, D. A. Strubbe, M. Jain, M. L. Cohen, and S. G. Louie, Comput. Phys. Commun. 183, 1269 (2012).

16 Excited-state properties 7/36 Materials: InSb, InAs, Ge, GaSb, Si, InP, GaAs, CdS, AlSb, AlAs, CdSe, CdTe, BP, SiC, C 60, GaP, AlP, ZnTe, ZnSe, c-gan, w-gan, InS, w-bn, c-bn, diamond, w-aln, LiCl, Theoretical Band Gap (ev) LDA GW Fluorite, LiF Experimental Band Gap (ev) Quasiparticle Gap Non-interacting Interacting Experiment Energy (ev) Optical absorption 2 1 S. G. Louie in Topics in Computational Materials Science, edited by C. Y. Fong (World Scientic, Singapore, 1997). 2 J. Deslippe, G. Samsonidze, D. A. Strubbe, M. Jain, M. L. Cohen, and S. G. Louie, Comput. Phys. Commun. 183, 1269 (2012).

17 The GW approximation 8/36

18 The GW approximation 9/36

19 Greens function 10/36 G(r t, r t ) = i Ψ 0 N T[ ˆψ(r t) ˆψ (r t )] Ψ 0 N Ψ 0 N the exact N electron ground state: Ĥ Ψ0 N = E0 N Ψ0 N the eld operator (Heisenberg picture): ˆψ(r t) = e iĥt ˆψ(r)e iĥt

20 Greens function 10/36 G(r t, r t ) = i Ψ 0 N T[ ˆψ(r t) ˆψ (r t )] Ψ 0 N Ψ 0 N the exact N electron ground state: Ĥ Ψ0 N = E0 N Ψ0 N the eld operator (Heisenberg picture): ˆψ(r t) = e iĥt ˆψ(r)e iĥt G(r t, r t ) = i Ψ 0 N T[ ˆψ(r t) ˆψ (r t )] Ψ 0 N

21 Greens function 10/36 G(r t, r t ) = i Ψ 0 N T[ ˆψ(r t) ˆψ (r t )] Ψ 0 N Ψ 0 N the exact N electron ground state: Ĥ Ψ0 N = E0 N Ψ0 N the eld operator (Heisenberg picture): ˆψ(r t) = e iĥt ˆψ(r)e iĥt G(r t, r t ) = i Ψ 0 N T[ ˆψ(r t) ˆψ (r t )] Ψ 0 N { i Ψ 0 = N ˆψ(r t) ˆψ (r t ) Ψ 0 N when t > t (electron) i Ψ 0 N ˆψ (r t ) ˆψ(r t) Ψ 0 N when t < t (hole)

22 Greens function 10/36 G(r t, r t ) = i Ψ 0 N T[ ˆψ(r t) ˆψ (r t )] Ψ 0 N Ψ 0 N the exact N electron ground state: Ĥ Ψ0 N = E0 N Ψ0 N the eld operator (Heisenberg picture): ˆψ(r t) = e iĥt ˆψ(r)e iĥt G(r t, r t ) = i Ψ 0 N T[ ˆψ(r t) ˆψ (r t )] Ψ 0 N { i Ψ 0 = N ˆψ(r t) ˆψ (r t ) Ψ 0 N when t > t (electron) i Ψ 0 N ˆψ (r t ) ˆψ(r t) Ψ 0 N when t < t (hole) = i Ψ 0 N ˆψ(r t) ˆψ (r t ) Ψ 0 N θ(t t ) + i Ψ 0 N ˆψ (r t ) ˆψ(r t) Ψ 0 N θ(t t)

23 Lehmann (Spectral) representation of Greens function 11/36 Insert the completeness relation: 1 = s,n Ψ s N Ψs N

24 Lehmann (Spectral) representation of Greens function 11/36 Insert the completeness relation: 1 = s,n Ψ s N Ψs N G(r t, r t ) = iθ(t t ) s + iθ(t t) s e i(e0 N Es N+1 )(t t ) g s (r)g s(r ) e i(e0 N Es N-1 )(t t) f s (r )f s (r) with quasiparticle amplitudes: g s (r) = Ψ 0 N ˆψ(r t) Ψ s N+1 f s (r) = Ψ s N-1 ˆψ(r t) Ψ 0 N

25 Lehmann (Spectral) representation of Greens function Insert the completeness relation: 1 = s,n Ψ s N Ψs N G(r t, r t ) = iθ(t t ) s e i(e0 N Es N+1 )(t t ) g s (r)g s(r ) + iθ(t t) s e i(e0 N Es N-1 )(t t) f s (r )f s (r) with quasiparticle amplitudes: g s (r) = Ψ 0 N ˆψ(r t) Ψ s N+1 f s (r) = Ψ s N-1 ˆψ(r t) Ψ 0 N Upon Fourier transforming: G(r, r ; ω) = g s (r)gs(r ) ω (E s s N+1 E0 N ) + iη + s f s (r)f s (r ) ω + (E s N-1 E0 N ) iη 11/36

26 Greens function Spectral function 12/36 Greens function contains spectral information on single-particle excitations i.e. changing the number of particles by one.

27 Greens function Spectral function 12/36 Greens function contains spectral information on single-particle excitations i.e. changing the number of particles by one. The poles of the Greens function give the corresponding excitation energies!

28 Greens function Spectral function 12/36 Greens function contains spectral information on single-particle excitations i.e. changing the number of particles by one. The poles of the Greens function give the corresponding excitation energies! Spectral function A(r, r ; ω) = 1 π ImG(r, r ; ω) = s g s (r)g s(r )δ(ω (E s N+1 E0 N )) + s f s (r)f s (r )δ(ω + (E s N-1 E0 N ))

29 Physical interpretation of the Greens function 13/36 For t > t, Ψ 0 N ˆψ(r t) ˆψ (r t ) Ψ 0 N probability amplitude that electron created at (r, t ) will go to (r, t) (r, t ) (r, t)

30 Physical interpretation of the Greens function 13/36 For t > t, Ψ 0 N ˆψ(r t) ˆψ (r t ) Ψ 0 N probability amplitude that electron created at (r, t ) will go to (r, t) (r, t ) (r, t) For t < t, Ψ 0 N ˆψ (r t ) ˆψ(r t) Ψ 0 N probability amplitude that hole created at (r, t) will go to (r, t ) (r, t ) (r, t)

31 Dyson equation 14/36 Many-body perturbation theory for calculating the Greens function.

32 Dyson equation 14/36 Many-body perturbation theory for calculating the Greens function. G 1 = G Σ G 1 DFT = G V xc

33 Dyson equation 14/36 Many-body perturbation theory for calculating the Greens function. G 1 = G Σ G 1 DFT = G V xc Equivalently one can solve: Quasiparticle equation: ( V ionic(r) + V Hartree (r) ) φ(r) + Kohn-Sham equation: Σ(r, r, E QP )φ(r ) = E QP φ(r) ( V ionic(r) + V Hartree (r) + V xc (r) ) ψ(r) = ɛ ψ(r)

34 Dyson equation 14/36 Many-body perturbation theory for calculating the Greens function. G 1 = G Σ G 1 DFT = G V xc Equivalently one can solve: Quasiparticle equation: ( V ionic(r) + V Hartree (r) ) φ(r) + Kohn-Sham equation: Σ(r, r, E QP )φ(r ) = E QP φ(r) ( V ionic(r) + V Hartree (r) + V xc (r) ) ψ(r) = ɛ ψ(r) All interactions are included in the Σ. Self energy is non-local and energy dependent

35 Dyson equation 14/36 Many-body perturbation theory for calculating the Greens function. Equivalently one can solve: Quasiparticle equation: ( 2 G 1 = G Σ G 1 DFT = G V xc What is this self energy Σ? 2 + V ionic(r) + V Hartree (r) ) φ(r) + Kohn-Sham equation: ( 2 Σ(r, r, E QP )φ(r ) = E QP φ(r) 2 + V ionic(r) + V Hartree (r) + V xc (r) ) ψ(r) = ɛ ψ(r) All interactions are included in the Σ. Self energy is non-local and energy dependent

36 The GW approximation /36

37 The GW approximation /36

38 The GW approximation 16/36 Hartree-Fock Approximation Σ x (r, r ) = ig(r, r ; t)v(r r ) occ = ψj (r)ψ j (r )v(r r ) j Σ = igv GW Approximation Σ GW (r, r ) = ig(r, r ; t)w(r, r, t) W: screened Coulomb interation Σ = igw W = ε 1 v Lars Hedin, Phys. Rev. 139, A796 (1965)

39 Photoemission process within Hartree-Fock 17/36 ω e Additional charge Independent particles. No relaxation of the system. Does not describe the physics in solids. Adapted from slides by M. Gatti

40 Photoemission process within GW 18/36 ω e Additional charge polarization and screening. Polarization from non-interacting electron-hole pairs (RPA) Classical interaction between additional charge and induced charge. Describes the physics well. Adapted from slides by M. Gatti

41 Quasiparticles 19/36 R. Mattuck, A Guide to Feynman Diagrams in the Many-Body Problem, McGraw-Hill Book Company, New York (1976)

42 Quasiparticles 19/36 Particle gets dressed! R. Mattuck, A Guide to Feynman Diagrams in the Many-Body Problem, McGraw-Hill Book Company, New York (1976)

43 Quasiparticles 19/36 Particle gets dressed! There is a polarization cloud around the particle which screens its interactions with the rest of the system. R. Mattuck, A Guide to Feynman Diagrams in the Many-Body Problem, McGraw-Hill Book Company, New York (1976)

44 Spectral function 20/36 A(ω) non-interacting ω ε i

45 Spectral function 20/36 A(ω) interacting non-interacting quasiparticle satellite ω ε i

46 Green function in the interaction representation 21/36 The self-energy may be evaluated by using Wick's theorem or by Schwinger's functional derivative method. Adapted from slides by F. Aryasetiawan

47 Green function in the interaction representation 21/36 The self-energy may be evaluated by using Wick's theorem or by Schwinger's functional derivative method. Introduce φ an external pertubation (set to zero at the end). Calculate variations of G with respect to φ. Adapted from slides by F. Aryasetiawan

48 Green function in the interaction representation 21/36 The self-energy may be evaluated by using Wick's theorem or by Schwinger's functional derivative method. Introduce φ an external pertubation (set to zero at the end). Calculate variations of G with respect to φ. ig(1, 2) = Ψ 0 T[Ŝ ˆψ(1) ˆψ (2)] Ψ 0 Ψ 0 Ŝ Ψ 0 Ŝ = T exp [ i d3 φ(3)ˆρ(3)] 1 r 1 t 1 Ψ 0 the exact ground state without perturbation: Ĥ Ψ 0 = E 0 Ψ 0 the eld operator (interaction picture) is indepenent of φ: ˆψ(rt) = e iĥt ˆψ(r)e iĥt Adapted from slides by F. Aryasetiawan

49 Green function in the interaction representation 21/36 The self-energy may be evaluated by using Wick's theorem or by Schwinger's functional derivative method. Introduce φ an external pertubation (set to zero at the end). Calculate variations of G with respect to φ. ig(1, 2) = Ψ 0 T[Ŝ ˆψ(1) ˆψ (2)] Ψ 0 Ψ 0 Ŝ Ψ 0 Ŝ = T exp [ i d3 φ(3)ˆρ(3)] 1 r 1 t 1 Ψ 0 the exact ground state without perturbation: Ĥ Ψ 0 = E 0 Ψ 0 the eld operator (interaction picture) is indepenent of φ: ˆψ(rt) = e iĥt ˆψ(r)e iĥt δŝ δφ(3) = it[ŝˆρ(3)] Adapted from slides by F. Aryasetiawan

50 Equation of motion 22/36 Equation of motion of the single particle Green function: ( i ) h 0 (r 1 ) G(1, 2) + i t 1 d3 v(1 3)G (2) (1, 2, 3, 3 + ) = δ(1 2) h 0 = V ext independent-particle Hamiltonian v(1 3) = v(r 1 r 3 )δ(t 1 t 3 ) G (2) (1, 2, 3, 4) = Ψ 0 T[Ŝ ˆψ(1) ˆψ(3) ˆψ (4) ˆψ (2)] Ψ 0 Ψ 0 Ŝ Ψ 0 two-particle Green Function Adapted from slides by F. Aryasetiawan

51 Equation of motion Equation of motion of the single particle Green function: ( i ) h 0 (r 1 ) G(1, 2) + i t 1 d3 v(1 3)G (2) (1, 2, 3, 3 + ) = δ(1 2) h 0 = V ext independent-particle Hamiltonian v(1 3) = v(r 1 r 3 )δ(t 1 t 3 ) G (2) (1, 2, 3, 4) = Ψ 0 T[Ŝ ˆψ(1) ˆψ(3) ˆψ (4) ˆψ (2)] Ψ 0 Ψ 0 Ŝ Ψ 0 Unfortunately heirarchy of equations: G(1, 2) G (2) (1, 2, 3, 4) G (2) (1, 2, 3, 4) G (3) (1, 2, 3, 4, 5, 6) two-particle Green Function... Adapted from slides by F. Aryasetiawan 22/36

52 Equation of motion 22/36 Equation of motion of the single particle Green function: ( i ) h 0 (r 1 ) G(1, 2) + i t 1 d3 v(1 3)G (2) (1, 2, 3, 3 + ) = δ(1 2) h 0 = V ext independent-particle Hamiltonian v(1 3) = v(r 1 r 3 )δ(t 1 t 3 ) G (2) (1, 2, 3, 4) = Ψ 0 T[Ŝ ˆψ(1) ˆψ(3) ˆψ (4) ˆψ (2)] Ψ 0 Ψ 0 Ŝ Ψ 0 i two-particle Green Function Dene the mass operator: d3 v(1 3)G (2) (1, 2, 3, 3 + ) d3 M(1, 3)G(3, 2) M = ivg (2) G 1 Adapted from slides by F. Aryasetiawan

53 Derivation of Hedin's equations δg(1, 2) δφ(3) = Adapted from slides by F. Aryasetiawan 23/36

54 Derivation of Hedin's equations δg(1, 2) δφ(3) δ Ψ 0 T[Ŝ = i ˆψ(1) ˆψ (2)] Ψ 0 δφ(3) Ψ 0 Ŝ Ψ 0 Adapted from slides by F. Aryasetiawan 23/36

55 Derivation of Hedin's equations δg(1, 2) δ Ψ 0 T[Ŝ = i ˆψ(1) ˆψ (2)] Ψ 0 δφ(3) δφ(3) Ψ 0 Ŝ Ψ 0 = Ψ 0 T[Ŝ ρ(3) ˆ ˆψ(1) ˆψ (2)] Ψ 0 Ψ 0 Ŝ Ψ + Ψ 0 T[Ŝ ˆψ(1) ˆψ (2)] Ψ 0 Ψ 0 T[Ŝˆρ(3)] Ψ 0 0 Ψ 0 Ŝ Ψ 0 2 Adapted from slides by F. Aryasetiawan 23/36

56 Derivation of Hedin's equations δg(1, 2) δ Ψ 0 T[Ŝ = i ˆψ(1) ˆψ (2)] Ψ 0 δφ(3) δφ(3) Ψ 0 Ŝ Ψ 0 = Ψ 0 T[Ŝ ρ(3) ˆ ˆψ(1) ˆψ (2)] Ψ 0 Ψ 0 Ŝ Ψ + Ψ 0 T[Ŝ ˆψ(1) ˆψ (2)] Ψ 0 Ψ 0 T[Ŝˆρ(3)] Ψ 0 0 Ψ 0 Ŝ Ψ 0 2 G (2) (1, 2, 3, 3 + ) = G(1, 2)ρ(3) δg(1, 2) δφ(3) Adapted from slides by F. Aryasetiawan 23/36

57 Derivation of Hedin's equations δg(1, 2) δ Ψ 0 T[Ŝ = i ˆψ(1) ˆψ (2)] Ψ 0 δφ(3) δφ(3) Ψ 0 Ŝ Ψ 0 = Ψ 0 T[Ŝ ρ(3) ˆ ˆψ(1) ˆψ (2)] Ψ 0 Ψ 0 Ŝ Ψ + Ψ 0 T[Ŝ ˆψ(1) ˆψ (2)] Ψ 0 Ψ 0 T[Ŝˆρ(3)] Ψ 0 0 Ψ 0 Ŝ Ψ 0 2 G (2) (1, 2, 3, 3 + ) = G(1, 2)ρ(3) But, δg δφ = GδG 1 δφ G δg(1, 2) δφ(3) Adapted from slides by F. Aryasetiawan 23/36

58 Derivation of Hedin's equations δg(1, 2) δ Ψ 0 T[Ŝ = i ˆψ(1) ˆψ (2)] Ψ 0 δφ(3) δφ(3) Ψ 0 Ŝ Ψ 0 = Ψ 0 T[Ŝ ρ(3) ˆ ˆψ(1) ˆψ (2)] Ψ 0 Ψ 0 Ŝ Ψ + Ψ 0 T[Ŝ ˆψ(1) ˆψ (2)] Ψ 0 Ψ 0 T[Ŝˆρ(3)] Ψ 0 0 Ψ 0 Ŝ Ψ 0 2 G (2) (1, 2, 3, 3 + ) = G(1, 2)ρ(3) But, δg δφ = GδG 1 δφ G δg(1, 2) δφ(3) M = ivg (2) G 1 = iv [ ρ δg δφ G 1] = V Hartree ivg δg 1 δφ Adapted from slides by F. Aryasetiawan 23/36

59 Derivation of Hedin's equations δg(1, 2) δ Ψ 0 T[Ŝ = i ˆψ(1) ˆψ (2)] Ψ 0 δφ(3) δφ(3) Ψ 0 Ŝ Ψ 0 = Ψ 0 T[Ŝ ρ(3) ˆ ˆψ(1) ˆψ (2)] Ψ 0 Ψ 0 Ŝ Ψ + Ψ 0 T[Ŝ ˆψ(1) ˆψ (2)] Ψ 0 Ψ 0 T[Ŝˆρ(3)] Ψ 0 0 Ψ 0 Ŝ Ψ 0 2 G (2) (1, 2, 3, 3 + ) = G(1, 2)ρ(3) But, δg δφ = GδG 1 δφ G δg(1, 2) δφ(3) M = ivg (2) G 1 = iv [ ρ δg δφ G 1] = V Hartree ivg δg 1 δφ Self Energy: Σ = M V Hartree = ivg δg 1 δφ Adapted from slides by F. Aryasetiawan 23/36

60 Derivation of Hedin's equations Σ(1, 2) = i d3d4 v(1 3)G(1, 4) δg 1 (4, 2) δφ(3) Adapted from slides by F. Aryasetiawan 24/36

61 Derivation of Hedin's equations Σ(1, 2) = i d3d4 v(1 3)G(1, 4) δg 1 (4, 2) δφ(3) v δg 1 δφ = v δv δg 1 δφ δv δg 1 = vɛ 1 δv = WδG 1 δv V = V Hartree + φ Adapted from slides by F. Aryasetiawan 24/36

62 Derivation of Hedin's equations Σ(1, 2) = i d3d4 v(1 3)G(1, 4) δg 1 (4, 2) δφ(3) v δg 1 δφ = v δv δg 1 δφ δv Σ(1, 2) = i δg 1 = vɛ 1 δv = WδG 1 δv d3d4 W(1, 3)G(1, 4) δg 1 (4, 2) δv(3) V = V Hartree + φ Adapted from slides by F. Aryasetiawan 24/36

63 Derivation of Hedin's equations Σ(1, 2) = i d3d4 v(1 3)G(1, 4) δg 1 (4, 2) δφ(3) v δg 1 δφ = v δv δg 1 δφ δv Σ(1, 2) = i Σ(1, 2) = i δg 1 = vɛ 1 δv = WδG 1 δv d3d4 W(1, 3)G(1, 4) δg 1 (4, 2) δv(3) d3d4 W(1, 3)G(1, 4)Γ(4, 2, 3) Vertex Γ(4, 2, 3) = δg 1 (4, 2) δv(3) V = V Hartree + φ Adapted from slides by F. Aryasetiawan 24/36

64 Derivation of Hedin's equations Σ(1, 2) = i d3d4 v(1 3)G(1, 4) δg 1 (4, 2) δφ(3) v δg 1 δφ = v δv δg 1 δφ δv Σ(1, 2) = i Σ(1, 2) = i δg 1 = vɛ 1 δv = WδG 1 δv d3d4 W(1, 3)G(1, 4) δg 1 (4, 2) δv(3) d3d4 W(1, 3)G(1, 4)Γ(4, 2, 3) Vertex Γ(4, 2, 3) = δg 1 (4, 2) δv(3) V = V Hartree + φ From equation of motion G 1 = i t h 0 V Hartree φ Σ = Γ = δg 1 δv = [ 1 + δσ ] δv Adapted from slides by F. Aryasetiawan 24/36

65 Derivation of Hedin's equations 25/36 Γ = 1 + δσ δv = 1 + δσ δg δg δv = 1 δσ δg GδG 1 δv G = 1 + δσ δg GΓG Adapted from slides by F. Aryasetiawan

66 Derivation of Hedin's equations 25/36 Γ = 1 + δσ δv = 1 + δσ δg δg δv = 1 δσ δg GδG 1 δv G = 1 + δσ δg GΓG Γ = 1 + δσ δg GGΓ Adapted from slides by F. Aryasetiawan

67 Derivation of Hedin's equations 25/36 Γ = 1 + δσ δv = 1 + δσ δg δg δv = 1 δσ δg GδG 1 δv G = 1 + δσ δg GΓG Polarization Γ = 1 + δσ δg GGΓ P = δρ δv = iδg δv = igδg 1 δv G = igγg Adapted from slides by F. Aryasetiawan

68 Derivation of Hedin's equations 25/36 Γ = 1 + δσ δv = 1 + δσ δg δg δv = 1 δσ δg GδG 1 δv G = 1 + δσ δg GΓG Polarization Γ = 1 + δσ δg GGΓ Screened interaction W = ɛ 1 v = δv δφ v = [ = 1 + v δρ δv δv δφ P = δρ δv = iδg δv = igδg 1 δv G = igγg [ 1 + δv Hartree δφ ] v = [ 1 + v δρ δv ɛ 1 ] v = [ 1 + v δρ ] v δφ ] v = v + vpw Adapted from slides by F. Aryasetiawan

69 Derivation of Hedin's equations 25/36 Γ = 1 + δσ δv = 1 + δσ δg δg δv = 1 δσ δg GδG 1 δv G = 1 + δσ δg GΓG Polarization Γ = 1 + δσ δg GGΓ Screened interaction W = ɛ 1 v = δv δφ v = [ = 1 + v δρ δv δv δφ P = δρ δv = iδg δv = igδg 1 δv G = igγg [ 1 + δv Hartree δφ ] v = W = v + vpw [ 1 + v δρ δv ɛ 1 ] v = [ 1 + v δρ ] v δφ ] v = v + vpw Adapted from slides by F. Aryasetiawan

70 Hedin's equations 26/36 Γ(123) = δ(12)δ(13) + G = G 0 + G 0 ΣG P(12) = i d(34) G(23)G(42)Γ(341) W = v + vpw Σ(12) = i d(34) G(14)W(1 + 3)Γ(423) d(4567) δσ(12) δg(45) G(46)G(47)Γ(673)

71 Hedin's equations 27/36 Start iteration Σ = 0 = G = G 0 Γ(123) = δ(12)δ(13) P 0 (12) = ig 0 (21)G 0 (12) = W 0 = v + vp 0 W 0 Σ(12) = ig(12)w(1 + 2) (Hartree Approximation) Can show that if you iterate then you will get the same diagrammatic expansion as with Wick's theorem. Wick's theorem diagrams are of the same order while here they come in dierent orders together.

72 Diagrammatic expansion in W 28/36 Lars Hedin, Phys. Rev. 139, A796 (1965)

73 Physical interpretation Coulomb hole + screened exchange 29/36 Σ =Σ SEX + Σ COH screened exchange + Couloumb hole occ. Σ SEX (r, r, ω) = ψ n (r)ψn(r )ReW(r, r, ω ɛ n ) all Σ COH (r, r, ω) = ψ n (r)ψn(r ) n n 0 dω ImW(r, r, ω ) ω ɛ n ω

74 Static limit Coulomb hole + screened exchange 30/36 Σ = Σ SEX + Σ COH Coulomb Hole (COH) Interaction of electrons with excitations (electron-hole, plasmons) i.e. screening charge Screened Exchange (SEX) Dynamically screened exchange (replacement v with W )

75 Static limit Coulomb hole + screened exchange 30/36 Σ = Σ SEX + Σ COH occ. Σ SEX (r, r ) = ψ n (r)ψn(r )W (r, r ) n Σ COH (r, r ) = 1 [ W (r, r ) v(r, r ) ] δ(r r ) 2 Coulomb Hole (COH) Interaction of electrons with excitations (electron-hole, plasmons) i.e. screening charge Screened Exchange (SEX) Dynamically screened exchange (replacement v with W )

76 Actual implementation 31/36 Quasiparticle equation: ( 2 2 +V ionic(r)+v Hartree (r) ) φ i (r)+ Σ(r, r, E QP )φ i (r ) = E QP φ i (r) Kohn-Sham equation: ( V ionic(r) + V Hartree (r) + V xc (r) ) ψ i (r) = E DFT ψ i (r)

77 Actual implementation 31/36 Quasiparticle equation: ( 2 2 +V ionic(r)+v Hartree (r) ) φ i (r)+ Σ(r, r, E QP )φ i (r ) = E QP φ i (r) Kohn-Sham equation: ( V ionic(r) + V Hartree (r) + V xc (r) ) ψ i (r) = E DFT ψ i (r) Assume φ i (r) ψ i (r)

78 Actual implementation 31/36 Quasiparticle equation: ( 2 2 +V ionic(r)+v Hartree (r) ) φ i (r)+ Σ(r, r, E QP )φ i (r ) = E QP φ i (r) Kohn-Sham equation: ( V ionic(r) + V Hartree (r) + V xc (r) ) ψ i (r) = E DFT ψ i (r) Assume φ i (r) ψ i (r) E QP = E DFT + ψ i (r) Σ V xc ψ i (r)

79 Actual implementation 32/36 Best G, Best W approach (H DFT 0 + Vxc DFT )ψ DFT = ɛ DFT ψ DFT

80 Actual implementation 32/36 Best G, Best W approach (H DFT 0 + Vxc DFT )ψ DFT = ɛ DFT ψ DFT χ 0 (12) = ig DFT (12)G DFT (21)

81 Actual implementation 32/36 Best G, Best W approach (H DFT 0 + Vxc DFT )ψ DFT = ɛ DFT ψ DFT χ 0 (12) = ig DFT (12)G DFT (21) Σ GW (12) = ig DFT (12)W DFT (1 + 2)

82 Actual implementation 32/36 Best G, Best W approach (H DFT 0 + Vxc DFT )ψ DFT = ɛ DFT ψ DFT χ 0 (12) = ig DFT (12)G DFT (21) Σ GW (12) = ig DFT (12)W DFT (1 + 2) ɛ QP = ɛ DFT + ψ DFT Σ GW V DFT xc ψ DFT

83 Excited-state properties molecules 33/36 F. J. Hüser, Ph.D Thesis, Technical University of Denmark

84 Excited-state properties solids 34/36 Materials: InSb, InAs, Ge, GaSb, Si, InP, GaAs, CdS, AlSb, AlAs, CdSe, CdTe, BP, SiC, C 60, GaP, AlP, ZnTe, ZnSe, c-gan, w-gan, InS, w-bn, c-bn, diamond, w-aln, LiCl, LDA Fluorite, LiF Experimental Band Gap (ev) Theoretical Band Gap (ev) Quasiparticle Gap Non-interacting Experiment Energy (ev) Optical absorption 2 1 S. G. Louie in Topics in Computational Materials Science, edited by C. Y. Fong (World Scientic, Singapore, 1997). 2 J. Deslippe, G. Samsonidze, D. A. Strubbe, M. Jain, M. L. Cohen, and S. G. Louie, Comput. Phys. Commun. 183, 1269 (2012).

85 Excited-state properties solids 34/36 Materials: InSb, InAs, Ge, GaSb, Si, InP, GaAs, CdS, AlSb, AlAs, CdSe, CdTe, BP, SiC, C 60, GaP, AlP, ZnTe, ZnSe, c-gan, w-gan, InS, w-bn, c-bn, diamond, w-aln, LiCl, Theoretical Band Gap (ev) LDA GW Fluorite, LiF Experimental Band Gap (ev) Quasiparticle Gap Non-interacting Interacting Experiment Energy (ev) Optical absorption 2 1 S. G. Louie in Topics in Computational Materials Science, edited by C. Y. Fong (World Scientic, Singapore, 1997). 2 J. Deslippe, G. Samsonidze, D. A. Strubbe, M. Jain, M. L. Cohen, and S. G. Louie, Comput. Phys. Commun. 183, 1269 (2012).

86 Excited-state properties Code 35/36 BerkeleyGW 1 J. Deslippe, G. Samsonidze, D. A. Strubbe, M. Jain, M. L. Cohen, and S. G. Louie, Comput. Phys. Commun. 183, 1269 (2012).

87 References 36/36 L. Hedin, Phys. Rev. 139 A796 (1965). L. Hedin and S. Lunqdvist, in Solid State Physics, Vol. 23 (Academic, New York, 1969), p. 1. F. Aryasetiawan and O. Gunnarsson, Rep. Prog. Phys (1998). W. G. Aulbur, L. Jonsson, and J. W. Wilkins, Sol. State Phys (2000). M. S. Hybertsen and S. G. Louie, Phys. Rev. B 34, 5390 (1986). F. Aryasetiawan presentation at ICTS meeting on Models to Materials 2014.